Discounts, Rationing, and Unemployment

Transcription

1 Discounts, Rationing, and Unemployment Alex Clymo December 3, 208 Abstract How are changes in discount rates transmitted to unemployment, and are they a quantitatively relevant driver of the Great Recession? In this paper I answer these questions in a search and matching model featuring endogenous capital accumulation. I show that endogenous capital amplifies the effects of a rise in discounts on unemployment. When firms make hiring decisions they must also jointly decide how much capital to provide their workers, which determines their labour productivity. Since capital is a long lived asset, a rise in discounts reduces investment, reducing the marginal product of labour and hence incentives to hire. I use the framework of Michaillat (202) to classify this effect as a rise in rationing unemployment. I estimate changes in discount rates during the Great Recession using data on investment, and find a modest but persistent rise in discounts. These discounts cause a 2.3pp rise in unemployment, mainly during the slow recovery, by causing 83% of the observed capital shallowing in the data. However, a decomposition exercise reveals that endogenous capital alone is not enough to allow discounts to explain the whole rise in unemployment during the recession. Keywords: discounts, unemployment, investment, great recession Department of Economics, University of Essex. a.clymo@essex.ac.uk. This paper started as a quick update of the third chapter of my PhD, before inevitably expanding into something quite different, and rather less quick. I thank my supervisor Wouter den Haan for invaluable support during the PhD. I have benefitted from comments on various iterations of this project from Sergio de Ferra, Axel Gottfries, Marcus Hagedorn, Andrea Lanteri, Kevin Sheedy, and seminar participants at the LSE, and EEA-ESEM conference 207. I thank the ESRC and Paul Woolley Centre for financial support during my PhD. All errors are my own.

2 Introduction What are the sources of the large and persistent disruption in the labour market that the US experienced during the Great Recession? Using the search and matching framework, recent work by Hall (207) and Kehoe et al. (forthcoming) proposes a rise in discounts as a potential explanation for reduced hiring incentives in recessions. In the presence of hiring frictions, firms hiring decisions are inherently forward looking: firms pay costs today to hire a worker, but receive the benefit of hiring them over the lifetime of the match. If firms discount these future benefits more (if discounts rise) this may reduce their incentives to hire, and hence lead to a rise in unemployment. While there are many reasons that discount rates could rise during a recession, one question the literature on discounts and unemployment has focused on is how a given rise in discounts is transmitted into unemployment. A key issue, going back to Mukoyama (2009) is that reasonable fluctuations in discount rates have small effects on unemployment in the standard search and matching model. One reason for this is that empirical job separation rates are very high in US data, meaning that when a firm hires a worker they do not expect them to stay very long. This implies that the effective total discount rate on firms hiring decisions is actually very high in a calibrated model, and hence changes in the pure discount rate can have limited effects on the incentive to hire. Both Hall (207) and Kehoe et al. (forthcoming) propose modifications of the standard model which amplify the effect of changes in discounts on unemployment, and in this paper I propose and test an additional amplification mechanism. The key idea of this paper is that when firms decide to hire an additional worker, they are not just deciding on whether to hire the worker herself, but also whether or not to invest in the capital required to make that worker productive. The basic search model (e.g. Pissarides, 2000) assumes single worker firms and linear production with labour as the only input. However, in reality if a firm wants to expand its workforce whilst maintaining their productivity, it must also invest in the machines and production lines, and in the extreme the entirely new plants, required for them to work with. Many search models do include capital accumulation, including early contributions by Merz (995), Andolfatto (996), and Pissarides (2000). However, this has previously be abstracted away from in search papers investigating the role of discounts. Capital accumulation and discounts interact in a very natural way, which could amplify the effect of discounts on unemployment. Suppose a firm expands its production by simultaneously hiring extra workers and investing in new capital. While these workers may turn over quickly due to high job separation rates, the installed capital which they work with has a much longer productive life. Hence this capital will be used to produce even with the workers who eventually replace those who have left. Put differently, the depreciation rate of capital is much lower than than the separation rate of workers. 2 This lower depreciation rate of capital means that capital is a longer lived investment, and should naturally be much more sensitive to changes in discounts. If a rise in discounts lowers investment, it will then also naturally reduce capital per worker, labour productivity, and the incentive to hire workers. If we interpret the amount of capital as a measure of the number of jobs or positions in a firm, then we can think of this as Kehoe et al. (forthcoming) show that in a model where wages are chosen via Nash bargaining over the total surplus, both the high job separation rate and the high job finding rate in US data play a role in increasing the effective discount rate of the firm s surplus from hiring. If wages are instead set as exogenous functions of productivity, then only the separation rate determines the effective discount rate, as I show in Section 2. 2 Standard values of the depreciation rate of capital are typically around 6.5% to 0% per year. Shimer (2005) calculates the worker separation rate to be 3.4% monthly, giving a 34% yearly separation rate. Thus the effective depreciation rate of workers is between 3.4 and 5.2 times higher than that of capital. 2

3 a distinction between worker and job flows. While individual worker flows are very high, the total number of jobs within a firm is empirically much more persistent 3 and capital is transferrable across workers even after an individual worker has left. In this paper I investigate whether, how, and by how much the inclusion of capital changes the propagation of discount rate shocks to unemployment. To this end, I build an otherwise standard search and matching model featuring endogenous capital accumulation, wage rigidity, and a time-varying discount rate. I make three main contributions. I first show using both analytical and numerical comparative statics results that endogenous capital provides a potentially powerful amplification mechanism for the effects of discounts on unemployment. Secondly, I follow Michaillat (202) and decompose the effect of discounts into frictional and rationing components. I show that discounts must raise rationing unemployment, while the effect on frictional unemployment is theoretically ambiguous. However, for sensible calibrations of the model frictional unemployment falls when discounts rise. Finally, I perform a quantitative decomposition exercise for the Great Recession. I show that, despite the amplifying role of capital, discounts have only modest effect on unemployment in my model, and mostly contribute to the slow recovery of unemployment rather than the initial peak. I begin my analysis with analytical results in the steady state of the model. I divide my results into two sections to highlight the additional role of endogenous capital in the model. I first investigate the effects of an increase in discounts on unemployment when capital is held constant. This model corresponds closely to the existing literature on discounts, with the exception that the model features diminishing marginal product of labour. I then move on to the full model, where capital is also allowed to adjust to a change in discounts according to the capital Euler equation. Using comparative statics results, I first show that a rise in discounts causes unemployment to increase when capital is fixed, since it causes a decrease in the discounted value of posting vacancies, as in the existing literature. I then show that a rise in discounts also increases unemployment when capital is allowed to adjust, and that this increase must be larger than when capital is fixed. Intuitively, unemployment increases more with endogenous capital because an additional effect operates: the rise in discounts lowers the incentive to invest in capital, which reduces the marginal product of labour and hence incentive to hire. Quantitative results from a calibrated version of the model confirm that these differences are large. Following Michaillat (202) I assume a wage rule linking wages to labour productivity with an empirically reasonable degree of wage rigidity. With this wage rule, doubling the discount rate has a negligible effect on unemployment with fixed capital, while the same exercise leads unemployment to triple when capital is endogenous. I then investigate the mechanisms through which discounts are transmitted to unemployment, and show that the transmission through capital works fundamentally differently from the effects in the linear search model. I apply Michaillat s (202) decomposition of unemployment into rationing and frictional components in my model, thus extending his analysis to a model with capital and variation in discounts. Michaillat defines rationing unemployment as the level of unemployment that would arise in a given model when hiring frictions are removed. Frictional unemployment is then defined as the extra unemployment arising due to the presence of hiring frictions. The combination of wage rigidity and diminishing marginal product of labour mean that in some states of the world rationing unemployment will be positive in my model. 3 See, for example, Elsby et al. (207) who show that many establishments exhibit zero net hiring despite nontrivial quit rates. 3

4 I first show that the small effect of discounts on unemployment when capital is fixed manifests only as a rise in frictional unemployment. I then show that the effects when capital is endogenous are more interesting. I show that when capital is free to adjust, rationing unemployment must rise when discounts rise. Rising discounts lowers capital intensity and hence the marginal product of labour. When wages are sticky this will cause unemployment to rise due to rationing in the labour market even in the absence of search frictions. However, I also show that the response of frictional unemployment is ambiguous, and it will also rise if wages are sufficiently flexible, while it will fall if wages are sufficiently sticky. Thus, whether the effect of discounts on unemployment is due to search frictions (frictional unemployment) or due to capital shallowing and sticky wages (rationing unemployment) is theoretically ambiguous. I thus move on to a calibrated version of the model, and show that for the baseline calibration, and indeed a wide range of calibrations, frictional unemployment falls when discounts rise. Thus, discounts appear to affect unemployment primarily as a rationing phenomenon. My final contribution is to perform decomposition exercises for the Great Recession to assess the importance of discounts in generating the observed patterns of unemployment during this period. I use data on investment to estimate the path of discounts. The discount rate estimated in this way rises modestly from an annualised rate of 5.4% pre-crisis to 6.% by the end of the sample. In contrast to discounts estimated from stock markets, these discounts rise gradually during the sample, but rise permanently. This reflects the fact that the capital-labour ratio in the US has permanently fallen by over 2% post-crisis, and the model requires a modest rise in discounts to explain this. I first feed this estimated rise in discounts through the model, applying it to the Euler equations for both hiring and investment. I solve the model non-linearly using a perfect foresight approach, and find that the rise in discounts generates a gradual but persistent rise in unemployment by 2.3 percentage points from 5% pre-crisis to 7.3% by the end of the sample. This modest rise in unemployment is driven entirely by a rise in rationing unemployment, and the rise in discounts causes a fall in frictional unemployment by the end of the sample. Intuitively, the estimated rise in discounts alone causes firms to reduce investment, which reduces the capital-labour ratio by 0%, a full 83% of the total decline in the data. This reduces the marginal product of labour, and since wages are sticky this causes a fall in hiring. Since the effects are concentrated towards the end of the recession, this suggests that rising discounts may be more important for explaining the slow recovery than the initial peak rise in unemployment. While I find that my model with endogenous capital amplifies the response of unemployment to a rise in discounts relative to a baseline model with exogenous capital, the overall effects are still modest. In the data, unemployment rose by 5 percentage points, reaching highs of 0% during the Great Recession. Additionally, if one considers the declines in labour participation over this period, I show that the effective model-consistent unemployment rate rises to over 3%. I thus perform a full decomposition exercise for the Great Recession where I decompose the path for unemployment over this period into components arising from five different shocks: discounts, TFP, job separations, labour force participation, and a residual shock to job creation. These shocks are estimated to exactly replicate the paths for capital, TFP, separations, participation, and employment in the data. I then feed these shocks through the non-linear model individually to assess each shock s contribution to the path of unemployment. I find that both rising discounts and declining TFP contribute to raising unemployment towards the end of the recession. Declining TFP growth contributes a rise of 2.pp to unemployment, 4

5 which is smaller but comparable to the 2.3pp contribution of rising discounts. Combined, the two forces represent a significant drag to unemployment during the recovery, and hence contribute to the slow recovery of unemployment. The effects of rising discounts start slightly sooner, but at unemployment s peak in 200 discounts are only contributing 0.8pp to the rise. The initial rise in unemployment is instead explained almost entirely by the residual shock to job creation incentives, which is unrelated to discounts in this model. Finally, I perform several exercises to investigate the role of discounts further, including a comparison of my discounts estimated from capital to discounts estimated from the stock market, as in Hall (207). I find that discounts estimated from stock markets rise much more at the beginning of the recession, but also recover quickly. This is in contrast to discounts estimated from capital, which rise less but more persistently. Interestingly, the estimated effects on unemployment are of a similar magnitude, but the effects and timings are drastically different: Discounts estimated from stocks increase unemployment at the beginning of the recession and operate via frictional unemployment, while discounts estimated from capital lead unemployment to rise at the end of the recession via rationing unemployment. Related Literature. This paper chiefly contributes to the literature on the role of discounts in driving unemployment over the business cycle. An early contribution by Mukoyama (2009) questioned the ability of discounts to meaningfully drive unemployment. More recently, two papers modify the standard model in order to generate larger movements in unemployment from smaller movements in discounts. Firstly, Hall (207) demonstrated that making wages stickier increased the power of variations in discounts. Rather than the Nash Bargaining assumption used in Mukoyama (2009), he uses the alternating offer model of Hall and Milgrom (2008) and shows that it is possible to calibrate the degree of wage stickiness to match the co-movement of discounts and unemployment. Apart from adding endogenous capital, my model differs from Hall s in two ways: ) I replace the alternating offer model of wages with an exogenous wage rule, with the degree of wage stickiness calibrated to match estimates from Haefke et al. (203). 2) I measure discounts from the Euler equation for capital investment, while Hall uses the stock market. I show that these two measures of discounts move quite differently during the Great Recession, leading to different implications for unemployment. Given that the stock market measures only a subset of firms while investment data covers the whole economy, these differences raise interesting questions about the heterogeneity of discount rates across firms. Secondly, Kehoe et al. (forthcoming) show that adding on-the-job human capital accumulation greatly amplifies the effects of a rise in discounts on unemployment. This is because human capital accumulation means that the surplus to a match grows over time as a worker becomes more productive, making the benefits to hiring felt further in the future. This makes the discounted benefits of hiring more sensitive to the discount rate. Making the benefits of a firm-worker match grow over time is similar in spirit to reducing the worker separation rate, which also increases the importance of returns further in the future. My model can be interpreted as giving another reason that firms hiring decisions should be more forward looking: When firms produce using both capital and labour, hiring decisions are intertwined with investment decisions. Since investment decisions naturally have long horizons due to the relatively low depreciation rates of capital, this spills over to firms hiring decisions. 5

6 In this paper I consider a simple representative-agent matching model where firms produce using both capital and employed workers. However, as discussed above the inclusion of long-lived capital naturally brings to mind the distinction between a firm hiring a worker versus firms opening a new job position. With multi-worker firms this distinction becomes meaningful because some hiring will be replacement hiring to replace workers who leave an existing job, while some hiring will be associated with firms expanding their total workforce, and hiring workers to fill new jobs. While individual workers transition very frequently across jobs, firms total employment levels are much more persistent. Elsby et al. (207) report that despite substantial turnover of workers, 40% of establishments report no change in net employment within one year, and 30% report no net change after three years. Thus, in contrast to the expected tenure of a single worker, net employment at establishments is much more persistent. Given this persistence, it is conceivable that changes in discount rates should have larger effects on firms net hiring decisions than implied by the high gross worker separation rates. In this paper, the persistence comes from the lower depreciation rate of capital, but it could also come from more detailed specifications of establishment dynamics and adjustment costs. By focusing on how variations in discounts affect firm investment, I also build on the large literature which investigates how financial frictions affect investment and economic activity. I interpret movements in these frictions through the lens of reduced form discounts affecting the incentive to invest in capital. Early contributions include Bernanke and Gertler (989), Kiyotaki and Moore (997), and Bernanke, Gertler, and Gilchrist (999). In this paper I consider the role of discounts in affecting investment in both capital and labour. Existing papers have built frameworks including financial frictions, capital, and labour market matching (for example, Christiano et al., 20, Mumtaz and Zanetti, 206) which could be reinterpreted through the lens of discounts. I investigate the importance of capital in amplifying discounts shocks in the search and matching model. In a similar spirit but without variations in discounts, den Haan et al. (202) argue that capital adjustment increases propagation in a search and matching model with endogenous job separations. Some papers interpret the vacancy posting cost in the search model as the cost of purchasing capital (e.g. Acemoglu and Shimer, 999). What is crucial for my analysis is the idea that capital is long lived and survives past the separation of any individual worker from the firm, which requires modelling the process for capital explicitly. Closest to my work are papers which consider discounts and frictions in both investment and hiring. Building on earlier work (Yashiv, 2000, Merz and Yashiv, 2007), Yashiv (206) estimates a model of joint adjustment costs in capital and labour. He uses a model-free forecasting VAR to relate hiring and investment to their expected future values. He finds that variations in both are driven mostly by their respective expected returns, thus finding an important role for variations in discounts. Interestingly, his estimation also finds that the rates of separation and depreciation do not play a meaningful role in job and investment values. This is supportive of the approach taken here and in Kehoe et al. (forthcoming) to find extensions of the search model to effectively make employment decisions more forward looking. Hall (206) extends the linear search and matching model of Hall (207) to include endogenous capital as well as several shocks. He uses this model to provide a decomposition of the Great Recession, similar in spirit to my exercise, but allowing for separate discounts to capital and labour. The most important difference from my paper is that Hall (206) does not provide a full model of the labour market, instead using a reduced-form representation of the labour market in which 6

7 unemployment can only depend on a product market wedge and labour discounts. Thus, the model does not allow the feedback from discount rates to capital to employment via the marginal product of capital, which is the central interest of this paper. The business cycle accounting approach of Chari et al. (2007) decomposes business cycles into constituent distortions in the equations of the real business cycle model. My time-varying discount rate is similar to the investment wedge in their model, but computed under the assumption of riskneutrality. They model the distortion in the labour market as a tax on labour income in an otherwise frictionless labour market, whereas I explicitly model search frictions and wage rigidity. Brinca et al. (206) update the approach to include data for many countries including the Great Recession, and find an important role for the investment wedge during this episode. By applying Michaillat s (202) decomposition of unemployment into rationing and frictional components to my model, I also thus extend his work to a model with endogenous capital and discounts as the shock driving unemployment. Michaillat s (202) model featured diminishing marginal product of labour and a fixed capital stock, making the aggregate production function decreasing returns to scale. I show that his classification continues to be relevant in a model with endogenous capital and constant returns at the aggregate level. Endogenous capital provides a new channel for rationing unemployment to exist, because in my model rationing unemployment can be caused by either low productivity or low capital, since both reduce the marginal product of labour. In Michaillat s (202) model a rise in discounts can only show up as an increase in frictional unemployment, while in my model it will initially manifest as frictional unemployment, but will manifest as rationing unemployment in the longer term as capital adjusts. As argued by Michaillat (202), the distinction between frictional and rationing unemployment is important because these two forms of unemployment have fundamentally different causes and hence different policy implications. Landais, Michaillat, and Saez (208a, b) use the rationing model to show that when recessions are driven by rationing, optimal unemployment insurance becomes more generous in recessions. This is because if unemployment is driven by rationing, and not by frictional forces, the costs of increasing unemployment insurance, for example that it reduces search effort by job seekers, become less relevant whenever rationing unemployment is high and frictional unemployment is low. The literature on discounts in search and matching models can also be understood as a reducedform way to represent the role of financial frictions in making hiring more expensive for firms. There is a large recent literature on this subject, with notable papers including Wasmer and Weil (2004), Petrosky-Nadeau (204), Petrosky-Nadeau and Wasmer (205), Quadrini and Sun (205), Schoefer (206), and Carrillo-Tudela et al. (208). Farmer (202) takes a different approach and removes the assumption of bargained wages, assuming a process for beliefs about the stock market which then drives job creation. The rest of the paper is structured as follows. In Section 2 I provide two simplified examples to illustrate the main ideas of the paper. In Section 3 I set up the model, and in Section 4 I perform analytical comparative statics. In Section 5 I perform numerical comparative statics, and in Section 6 I analyse the Great Recession using numerical methods. In Section 7 I provide additional results, and in Section 8 I conclude. 7

8 2 Two Motivating Examples In this section I present two stylised models to illustrate the key ideas in this paper. I first present a linear search and matching model in the spirit of Hall (207) and Kehoe et al. (forthcoming), and discuss the features determining the strength of the effect of discounts on unemployment. I then present a model without search frictions but with capital and unemployment driven by sticky wages, and discuss the transmission of discounts in this model. In both models, I take the extreme case of a fully sticky real wage, which does not respond at all to discounts. This makes the mechanisms transparent and simplifies the exposition. I relax this assumption in the later parts of the paper, where I use an empirically calibrated degree of stickiness. 2. Basic search model Consider the steady state of a discrete-time linear search model with single-worker firms. There is a unit mass of workers. Matches produce constant output z and separate with exogenous probability ρ. Workers are paid a constant exogenous wage w, and firms post vacancies at flow cost κ. Market tightness, θ v/u is the ratio of vacancies, v, to unemployment, u. Vacancies are filled with probability q(θ) coming from a constant returns to scale matching technology with q (θ) < 0. Unemployed workers find jobs at rate f(θ) = q(θ)θ and steady state unemployment is given by u = ρ/(f(θ) + ρ). Firms discount the future at rate r per period. In equilibrium, the level of tightness is such that the free entry condition equates the value to the firm of a match to the expected cost of filling a vacancy: κ q(θ) = z w () r + ρ This equation gives the intuition for how a rise in discounts increases unemployment. For a fixed wage, the firm discounts the per-period profit z w at total rate r+ρ capturing both time discounting and the probability that the match will dissolve each period. A rise in the time discount rate ( r) increases the total discount and hence the value of the match to the firm. Thus, market tightness must decrease ( θ) in order to raise the probability that vacancies are filled ( q(θ)) to reduce the expected vacancy posting cost and restore the free entry condition. While the intuition is clear, the quantitative importance of this channel is not yet established. To do so, assume a Cobb Douglas matching function such that the vacancy filling probability is given by q(θ) = ψ 0 θ ψ, where ψ 0 controls match efficiency and ψ controls the elasticity of matches with respect to unemployment. Plugging this into () and rearranging yields an expression for tightness: θ = ( ψ0 κ ) z w ψ (2) r + ρ Consider comparative statics in this equation across values of the discount. While it looks like all the parameters of the model control the response of tightness to discounts, the assumption of a fully fixed wage means that the result is actually much simpler. Suppose the model is calibrated with a steady state discount r and values of the other parameters leading to steady state tightness θ. To find the level of tightness as we vary r take the ratio of (2) for a generic r versus the calibrated level r : θ θ = ( r ) + ρ ψ (3) r + ρ 8

9 This shows us that across all calibrations of the model featuring the same steady-state tightness, the responsiveness of tightness to the discount rate is controlled only by the steady state discount, r, the rate of job separations, ρ, and the matching elasticity, ψ. Thus, features of the calibration such as the fraction of the surplus accruing to the firm (z w) are actually irrelevant to how powerful discounts are, if these models are calibrated to match the same steady state level of tightness. 4 With this knowledge in hand, I discuss the power of discounts in a standard calibration of the model. I use a monthly calibration, and I take the monthly job separations rate to be ρ = , implying a yearly rate of 38%. This is the value used in my full calibration, taken as the total separation rate to both unemployment and non-participation from the 2008M CPS data. I choose a steady state discount rate of r = , which corresponds to a yearly rate of 5.37%. This rate is relatively standard, and is calibrated from pre-crisis data in my full model, as detailed in Section 5. I use a standard matching elasticity of ψ = 0.5. Following Shimer (2005), with this matching function I can normalise θ =, and I choose match efficiency to set steady state unemployment to u = As discussed above, the remaining details of the calibration are irrelevant and are omitted. I present the results of a simple experiment in Table. In each column from left to right I give the steady state level of unemployment as the discount rate is increased from its calibrated value up to a maximum value of r = 2r. The first row gives the results for the baseline calibration, and the remaining rows give the results for various alternative calibrations. Table : Effect of doubling discounts on unemployment in the basic search model r.25r.5r.75r 2r Baseline ρ ρ = ρ = ρ = r = Notes: Columns give steady state unemployment when discount rate, r, is raised to given multiples of initial calibrated value, r. Top row gives results for baseline calibration. Second row gives results for calibration with near-permanent jobs (ρ = ), third row for separation rate half of baseline, fourth for double baseline, fifth row for jobs that last one month only (ρ = ), and final row for calibration with calibrated discount rate of r = 0.05 per month. I choose to investigate the effects of doubling discounts because this implies large movements in asset prices. For example, in the absence of other factors, permanently doubling discounts should exactly half the price-dividend ratio in a Gordon Growth model. Hall (207, Figure 8) reports a similar decline in the S&P500 index during the Great Recession, which I replicate in Figure 7, so this provides a natural benchmark. Additionally, this corresponds roughly to the increase in discounts considered by Hall (207) during the Great Recession. 5 4 This stands in contrast to the comparative static with respect to productivity, z. Here the level of the surplus is important because it will determine how large the proportional change in z w is for a given proportional change in z. The proportional change in z w is larger when the firm s surplus is smaller. This issue is not important for the powerfulness of discounts when the wage is completely fixed. 5 In Table 3 he reports that monthly discounts increase by slightly more than double, 233%, going from the average 9

10 Despite the large increase in discounts, the first row of Table shows that this has minimal effects on unemployment in the baseline calibration. Going from the first to last columns shows the effects of doubling discounts from the calibrated value, which only raises unemployment by 0.5 percentage points. This is even under the relatively extreme assumption that wages are fully fixed, and do not fall at all to offset the rise in unemployment. Studying (3) gives the intuition for this result. The proportional response of tightness to a change in discounts depends not on the change in the discount rate itself, but on the proportional change in the total effective discount rate, including the job separation rate: r + ρ. In the baseline calibration the separation rate, which is taken directly from the data, is an order of magnitude larger than the time discount rate: ρ = versus r = Hence increasing the time discount rate will have a small impact on the effective discount rate, since this is dominated by the job separation rate. Since ρ is so large, increasing r from r by 00% only increases r + ρ by 9.6%. In other words, with such a high job separation rate, jobs simply don t last very long on average. Hence hiring decisions are not too far from being static, and increasing the discount rate can have only a limited impact on hiring decisions. To further demonstrate this effect, in rows 2 to 5 of Table I give the results of the same exercise for different values of ρ. In row 2 I let ρ approach zero, meaning that jobs become near-permanent. Now the effects of an increase in discounts are much larger: doubling discounts nearly doubles the unemployment rate. In row 5 I increase the separation rate to ρ =, meaning that all jobs last exactly one month. In this case the effect of raising discounts effectively disappears. However, for reasonable perturbations of the separation rate from its empirical counterpart the effects do not change dramatically from the baseline, as I show in rows 3 and 4 where I half and double the separation rate respectively. In row 6 I show that the effects of doubling discounts are also larger if I recalibrate the baseline discount rate to be very large. If I set r = 0.05 (implying a yearly discount rate of nearly 80%) then r is of the same order of magnitude as ρ, and so small changes in r can start to have meaningful effects on the overall discount, r + ρ. Thus, it is not the high separation rate itself which makes discounts weak at moving unemployment, but rather the fact that it is so much higher than the discount rate. Having established that the basic linear search model with fully rigid wages is not able to generate meaningful permanent movements in unemployment from large permanent changes in discounts, I now discuss how Hall (207) and Kehoe et al. (forthcoming) modify the model to amplify the effects. The fact that Hall (207) finds larger effects is initially surprising, given that the only modification he makes to the standard model is to add wage rigidity. Since I take the limit of fully rigid wages, it might seem that I have made the most extreme possible assumption, and that the small 0.5pp increase in unemployment that I find is the maximum that can be found by increasing wage rigidity. However, Hall s (207) wage setting protocols actually lead to the real wage increasing when discounts increase. This behaviour can be rationalised under Hall and Milgrom s (2008) alternative offer bargaining. Following a rise in discounts, the real wage also rises, which reduces the share of the surplus accruing to the firm and hence further depresses the incentive to create jobs. Thus, part of Hall s (207) insight is that it is necessary for the firm s share of the surplus to shrink in order to meaningfully amplify the effect of discounts on unemployment. state to the state of the world corresponding to the Great Recession in his model. Due to the large effect on stock prices of a permanent doubling of discounts, I will refer to this increase in discounts as large. When considering dynamics the interpretation of a large change in discounts is modified, as I discuss further at the end of the section. 0

11 Kehoe et al. (forthcoming) take a different approach to amplify the effect of discounts. They build a model where workers accumulate human capital on the job. When a worker is hired, their human capital grows over time, allowing them to produce more output over time. This makes matches more forward looking, since more of the surplus from the match will now be produced further in the future, making the discounted value of the match more sensitive to discounts. In terms of the simple model presented above, this can be understood as reducing the effective job separation rate, which I showed increased the power of discounts. 6 Overall, this section shows us two things. Firstly, the basic linear search and matching model generates small permanent movements in unemployment from permanent changes in discounts even in the limit of fully rigid wages. Secondly, extensions to the model can amplify the role of discounts either by changing wage-setting protocols to reduce the firm s share of the surplus when discounts rise (Hall, 207), or by making matches effectively longer lived (Kehoe et al., forthcoming). One caveat to the above results is that they consider a permanent increase in discounts, while in reality recessions and associated falls in stock prices are temporary. A much larger temporary rise in discounts is needed to generate a temporary 50% fall in stock prices than the permanent rise in discounts required to generate a permanent 50% fall. Thus, considering the dynamics of discounts has the potential to amplify their effects on unemployment. In subsection 7.6 I explicitly compute the path of discounts required to match the path of stock prices during the recession, and find that it does have a meaningful but short lived effect on unemployment. However, the required rise in discounts is found to be very large, around a factor of ten, consistent with the results of this section that it takes large movements in discounts to move unemployment in the basic search model. 2.2 Pure rationing model Consider now a model without search frictions but where output is produced using both labour and capital. The production function is y = zk α l α, where z now denotes total factor productivity (TFP). I abstract from the hours-per-worker margin, and let l denote employment and u = l unemployment. I consider a simple model of unemployment where jobs are rationed due to the real wage being stuck above the market-clearing level. For a given wage, firms hire until the marginal product of labour (MPL) equals the real wage. However, I assume the real wage is fixed, and that workers are off their labour supply curves. Without hiring frictions, the firm s optimality condition for labour is the standard static condition equating with wage with the MPL: ( α)zk α l α = w (4) For a given wage, TFP, and level of capital, we can solve for equilibrium employment and unemployment as: ( ) ( ) ( α)z α ( α)z α l = k = u = k, (5) w w where I have assumed for simplicity that parameters are such that unemployment is always nonnegative. The expression for l in (5) shows that in this model firms set the level of employment 6 To see this, suppose that workers initially produce z and are paid w when hired. These both grow at rate g per period as workers accumulate human capital. The discounted value to the firm of a new match is now J = (z w) ( + β( ρ)( + g) + β 2 ( ρ) 2 ( + g) ) showing that increasing the growth rate of productivity ( g) is equivalent to reducing the separations rate ( ρ).

12 optimally proportional to the level of capital. Intuitively, firms hire workers until the marginal product of labour falls to equal the real wage. Reducing the level of capital makes workers less productive and reduces the MPL, and firms respond by reducing hiring by the same proportion to restore the MPL to the real wage. How does an increase in discounts affect unemployment in this model? If TFP and the real wage are fixed, then it has to be through the level of capital. Just as we expect increased discounts to reduce desired hiring when hiring is frictional, they will also reduce desired investment and capital. An important question is how powerful a channel this is for affecting unemployment. Clearly this will depend on how much capital moves during a recession. To give a sense of the potential quantitative magnitudes, consider a simple exercise. Suppose that unemployment is initially 5%, so that initial employment is l = In order to increase the unemployment rate to 0% we need employment to fall to l = 0.9, which is a 5.3% reduction from its initial value. Since labour moves proportionally to capital in this simple example, it would only take a 5.3% reduction in capital to push unemployment up to 0%. As I show in the data used for the decomposition exercises of Section 6, in the US detrended capital had fallen by 4.5% from 2008 to 200, and by more than 5% by 208. Hence, movements in capital in the Great Recession are large enough to potentially contribute meaningfully to unemployment. Of course, wages are not fully rigid in reality, which will temper this mechanism, and so incorporating a realistic degree of wage flexibility is important for determining how powerful an effect declines in investment can have on unemployment. Additionally, firms also face matching frictions when hiring, as in the model of the first example. Since these frictions fall in recessions because unemployed workers are abundant, this will also encourage hiring, and limit the ability of declines in capital to increase unemployment. Thus, in order to formally test whether endogenous capital accumulation can meaningfully amplify the effect of discounts on unemployment, in the remainder of the paper I build a full model which blends and extends the mechanisms highlighted in these two simple examples. 3 Model In this section I set up the baseline model to be used throughout the rest of the paper. The model is an extension of the standard search and matching model to include capital, sticky wages, and time-varying discounts. Time is discrete and the horizon is infinite. For simplicity, I restrict myself to deterministic economies. There are two agents in the model: a representative (multi-worker) firm, and a representative household. The assumption of time-varying discounts is a stand in for any financial or contracting frictions, or preferences, which induce risk-adjusted time preferences to change. Rather than spell these out, I follow Hall (207) and simply assume that the representative household s discount factor, β t, is time varying. β t describes the discount applied to real consumption between t and t +, and it follows a deterministic sequence {β t } t=0. Since variations in this rate stand in for all risk-adjusted time preferences, I assume that agents are risk neutral, and markets in the model are complete so that I do not have to consider the firm s financial structure. 2

13 3. Firms The representative firm produces output using a Cobb-Douglas production function y t = z t kt α l α t, featuring one period time-to-build in both capital and labour. There is no intensive margin for labour, so l t refers to employment. z t is a common productivity shock, which follows a deterministic sequence {z t } t=0. The firm hires labour subject to a standard matching friction. At time t the firm posts v t vacancies at flow-cost κ per vacancy. The firm takes the vacancy-filling probability q t as given. I assume that the law of large numbers holds, so the firm receives q t v t new employees with certainty, who produce from the next period. Workers separate from the firm at the exogenous rate ρ, and the firm s labour stock thus evolves as The firm additionally accumulates a stock of capital according to l t = q t v t + ( ρ)l t. (6) k t = i t + ( δ)k t, (7) where i t is investment and δ is proportional depreciation. I assume that adjustment costs in capital are external to the firm, so that capital trades at a price p k t which the firm takes as given. The price is given by the weakly increasing function p k t = p k (i t /k t ), which can be derived as the solution to the profit maximisation problem of a competitive capital goods producing sector with convex adjustment costs. Let variables without time subscripts denote steady state values. I normalise the price of capital to one in steady state, by noting that i/k = δ in steady state, and assuming that p k (δ) =. Cashflow, e t, is given by output less wages, vacancy posting costs, and investment: e t = z t kt α l α t w tl t p k t i t κv t. Plugging in the equations for the evolution of capital and labour, (6) and (7), gives cashflow in terms of the stock variables: e t = z t k α t l α t w tl t p k t (k t ( δ)k t ) κ q t (l t ( ρ)l t ). (8) The firm maximises the discounted sum of cashflows, discounted by the households time-varying discount factor, β t. Letting β =, formally, at time 0 they maximise ( ) t=0 Π t j=0 β j e t subject to the sequence of constraints (6), (7), and (8). The solution to this problem is characterised by a pair of Euler equations for capital and labour respectively: ( ) p k t = β t αz t+ kt α lt α + p k t+( δ) (9) ( κ = β t ( α)z t+ kt α lt α w t+ + q t ) ( ρ)κ q t+ For both assets, the firm trades off the adjustment cost of acquiring one more unit of the asset today with the discounted benefit of the marginal cashflow generated tomorrow, and the adjustment cost saved tomorrow. Search frictions thus naturally act as a labour hiring cost, with it costing the firm κ q t to acquire one more worker, just as it costs p k t to acquire one more unit of capital. Due to the Cobb Douglas assumption, the MPL is simply equal to a constant fraction of measured (0) 3

14 labour productivity: ( α)z t+ k α t l α t = ( α)y t+ /l t. 7 The worker separation rate, ρ, acts as a depreciation rate, symmetrically to the capital depreciation rate, δ, in the firm s hiring decision. 3.2 Labour market and matching technology I assume a standard constant returns to scale (CRS) matching function for creating new employment matches. This takes the stock of unemployed workers, defined as u t, and posted vacancies, v t, to create m t = m(u t, v t ) new matches. Market tightness is defined as θ t v t /u t. Given the CRS assumption, this gives the vacancy filling rate as a function q t = q(θ t ) m(θt, ). The job finding rate, f t, is given by f t = f(θ t ) m(, θ t ). I maintain as an assumption throughout this paper that the matching function is such that the job finding rate is strictly increasing in tightness. I initially assume that the labour force is constant at mass one. This gives unemployment as simply u t = l t. 8 In the decomposition exercise for the Great Recession in Section 6 I allow the participation rate to change over time, to capture the large movements in participation and demographics seen over that period. 3.3 Wage rule Following Michaillat (202), I specify wages in my model in terms of an exogenous wage rule, rather than as the outcome of an endogenous bargaining process. This allows me to directly control the degree of wage rigidity, which is an important object in determining how discount rate shocks are transmitted to the economy. In particular, consider the market clearing wage in this model, in the absence of matching frictions. Since the labour force is fixed at one, the market clearing wage is simply the marginal product of labour when all workers are employed (l t = ) and for a given level of capital and technology: = ( α)z t kt α. Recalling that steady-state values of a variable are denoted by an absence of subscripts, the steady state wage is given by w. Note that we must have the steady state wage being below the MPL in steady state (w < w mc = ( α)zk α ) to compensate the firm for vacancy posting costs. I assume that the wage rule is given by w mc t w t = ω ( z t k α t ) γ, () with ω w(zk α ) γ controlling the average level of the wage. In response to changes in either z t or k t, the wage stickiness parameter γ controls how far the wage is able to adjust towards the new market clearing wage. When γ = 0 the wage is fully rigid and fixed at the steady state wage: w t = w. When γ = the wage is flexible, and adjusts one-for-one with changes in the market clearing wage: w t = wt mc (w/w mc ). Intermediate values of γ give partial wage flexibility. Recent work emphasises that the flexibility of wages in existing matches versus new jobs is different, with wages in new matches responding much more to changes in labour productivity (e.g. Haefke et al., 203). I do not make the distinction between existing and new wages in my model. Accordingly, when it comes to numerical work I will calibrate wage stickiness to the level of new matches to be conservative. 7 Iterating forward on (0) yields the standard free entry condition modified to include capital in production and time-varying discounts. At time 0 we have κ q 0 = ( ) t=0 Π t j=0 β j (( α)yt+/l t w t+) and similar for t > 0. 8 For simplicity of exposition, I suppose that unemployment is always positive. 4